What if you were told that a triangle has sides that measure 3, 4, and 5? How could you determine which of the triangle's angles is largest? Smallest? After completing this Concept, you'll be able to use triangle theorems to solve problems like this one.

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Guidance

Look at the triangle below. The sides of the triangle are given. Can you determine which angle is the largest? The largest angle will be opposite 18 because that is the longest side. Similarly, the smallest angle will be opposite 7, which is the shortest side.

This idea is actually a theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

The converse is also true: If one angle in a triangle is larger than another angle in that triangle, then the side opposite the larger angle will be longer than the side opposite the smaller angle.

We can extend this idea into two theorems that help us compare sides and angles in
two
triangles If we have two congruent triangles
and
, marked below:

Therefore, if
,
, and
, then
.

Now, let’s make
. Would that make
? Yes. This idea is called the
SAS Inequality Theorem
.

The SAS Inequality Theorem:
If two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.

If
and
, then
.

If we know the third sides as opposed to the angles, the opposite idea is also true and is called the
SSS Inequality Theorem
.

SSS Inequality Theorem:
If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle's two congruent sides is greater in measure than the included angle of the second triangle's two congruent sides.

Vocabulary

Language:

The SAS Inequality Theorem states that if two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.

SSS Inequality Theorem

SSS Inequality Theorem

The SSS Inequality Theorem states that if two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle's two congruent sides is greater in measure than the included angle of the second triangle's two congruent sides.

Triangle Sum Theorem

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

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Description

Compare sides and angles in triangles.

Learning Objectives

Here you'll learn how to order the angles of a triangle from largest to smallest based on the length of their opposite sides. You'll also learn the SAS and SSS Inequality Theorems.